Mixed Tensor - Changing The Tensor Type

Changing The Tensor Type

Consider the following octet of related tensors:

$T_{\alpha \beta \gamma}, \ T_{\alpha \beta} {}^\gamma, \ T_\alpha {}^\beta {}_\gamma, \ T_\alpha {}^{\beta \gamma}, \ T^\alpha {}_{\beta \gamma}, \ T^\alpha {}_\beta {}^\gamma, \ T^{\alpha \beta} {}_\gamma, \ T^{\alpha \beta \gamma}$ .

The first one is covariant, the last one contravariant, and the remaining ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor can be lowered using the metric tensor g_μν, and a given covariant index can be raised using the inverse metric tensor gμν. Thus, g_μν could be called the index lowering operator and gμν the index raising operator.

Generally, the covariant metric tensor, contracted with a tensor of type (M, N), yields a tensor of type (M − 1, N + 1), whereas its contravariant inverse, contracted with a tensor of type (M, N), yields a tensor of type (M + 1, N − 1).

Read more about this topic: Mixed Tensor

Famous quotes containing the words changing the, changing and/or type:

“A love affair should always be a honeymoon. And the only way to make sure of that is to keep changing the man; for the same man can never keep it up.”
—George Bernard Shaw (1856–1950)

“Anything in history or nature that can be described as changing steadily can be seen as heading toward catastrophe.”
—Susan Sontag (b. 1933)

“The ideal American type is perfectly expressed by the Protestant, individualist, anti-conformist, and this is the type that is in the process of disappearing. In reality there are few left.”
—Orson Welles (1915–1984)